This invention relates to a pendulum, and more particularly, to a new and improved pendulum arm in the form of a flexure which is made of energy-conserving material, such as quartz, and which has a structure that is capable of reproduction in multiple substantially identical units, all of which exhibit substantially identical length, flex, and resonant operating characteristics. Further still, the present invention relates to a new and improved method of construction of such a pendulum arm flexure.
A pendulum is formed by a mass or “bob” that is connected to one end of a pendulum arm. The other end of the pendulum arm is pivotally connected to a stationary structure at a point of suspension or a center of motion. Energy imparted to the bob causes it to swing back and forth in an arc of oscillation at the point of suspension. Gravity sustains the oscillation of the bob until friction dissipates the oscillation energy of the swinging bob.
The time required for the pendulum bob to swing from one maximum amplitude end point in the arc of oscillation back to that same point is the period (T) of the swing. The period (T) of the swing, the gravity (g) and the length of the pendulum arm (L) are related to one another in an ideal pendulum by the following equation (1):T=2π[L/g]1/2  (1)Knowing or measuring two of the three variables length (L), gravity (g) or period (T) permits the other variable to be calculated. In this manner, a pendulum may be used as a measurement device for determining gravity (g), or precise time intervals (T), or the frequency (f) of the oscillation of the pendulum. The period (T) and the frequency (f) are inversely related to one another by the following well known equation (2):f=1/T  (2)
It is desirable to minimize the oscillation energy loss associated with the swinging pendulum. Oscillation energy losses have the effect of changing the period (T) and/or increasing the frequency (f). A changing period (T) or frequency (f) makes it very difficult to calculate with precision the quantity which is to be measured with the pendulum. Adding energy to replace that energy lost to friction is very difficult in a pendulum, because the added energy may create aberrations in the swing of the pendulum which in turn affect the ability to precisely measure the desired variable. While energy loss in a pendulum cannot be avoided altogether, minimizing the energy loss has the effect of enhancing the accuracy of measurement.
One significant source of energy loss in a pendulum is the friction at the point of suspension where the pendulum arm connects to the stationary structure. The friction from the movement of the pendulum arm relative to the stationary structure dissipates energy. Even a knife-edge point of suspension creates enough friction to adversely affect the period (T) and frequency (f) in a precision pendulum.
One known technique of diminishing energy loss at the point of suspension is to prevent the pendulum arm from moving relative to the stationary structure. To do so, the pendulum arm must be formed as a resilient flexure which is rigidly connected to the stationary structure at the point of suspension. The other end of the flexure is rigidly connected to the pendulum bob. The rigidly connected ends of the flexure do not move relative to the objects to which they are connected, so there is no frictional loss associated with relative movement at these points. Instead, the flexure bends back and forth as the bob swings in its arc of oscillation.
One known pendulum flexure is formed from a resilient, energy conserving material, such as quartz (fused silica) or other similar amorphous material. Flexing the material in one direction temporarily stores energy as intermolecular or van der Waals forces within the resilient material of the flexure. When the flexure flexes in the opposite direction, the stored energy is released. In this manner, a significant quantity of the oscillation energy is preserved, minimizing the loss of oscillation compared to the frictional losses from relative mechanical movement.
The known pendulum arm flexure is formed of quartz or other energy-conserving material. Examples are described in two theses: A Pendulum Gravimeter for Measurement of Periodic Annual Variations in the Gravitational Constant, by William F. Hoffman, Princeton University, Jan. 1962; and A Pendulum Gravimeter for Precision Detection of Scalar Gravitational Radiation, by David R. Curott, Princeton University, May 1965. The quartz pendulum arm flexures described in these theses are formed by heating the center section of a solid quartz rod until it achieves a viscous and flowable state, and then stretching the viscous center section to draw it out to a long, small diameter fiber extending between the larger unchanged ends of the rod. The rod transitions or necks down from the full diameter ends to the small diameter center fiber. The transitions occur in an unpredictable manner according to the uniformity of heat distribution in the center section of the quartz rod, the amount of heat energy in the center section prior to stretching, the rate at which the solid rod is stretched, and the viscosity of the heated center portion from which the fiber is formed, among other variables. The fiber itself is not of a uniform diameter, because the stretching occurs in an uncontrolled manner. The necked down transition portions between the full diameter ends of the rod and the center fiber are also variable in characteristics, due to the transitions occurring in an uncontrolled manner.
As a consequence of these uncontrolled variables, the length (L) of the pendulum arm is not predictable, and the flex characteristics of the flexure are also unpredictable. The necked down transition portions do not precisely demarcate points which establish the length (L) of the fiber which forms the pendulum arm. The thinnest portions of the necked down transition portions adjacent to the fiber may flex slightly along with the fiber, thereby varying the length (L) of the pendulum arm. Furthermore, the nonuniform diameter or thickness of the fiber itself will have different flexure characteristics.
These idiosyncratic aspects of known prior art quartz pendulum arm flexures are not of principal concern in those pendulum devices which utilize only a single pendulum supported by a single flexure. The operating characteristics of the pendulum device are adapted to the unique characteristics of the single flexure. However, in pendulum devices which require two flexures to support a single bob, or in pendulum devices which use two separate pendulums operating at the same oscillation frequency, it is important that multiple pendulum arm flexures have substantially the same length, flex and resonant operating characteristics. Pendulum arm flexures having substantially the same length, flex and resonant operating characteristics achieve predictable oscillatory behavior. Using pendulum arm flexures which have significantly different length, flex and resonant operating characteristics result in undesirable modes of movement of a single pendulum supported by two flexures. The undesirable modes of movement consume additional energy and adversely affect the desired operation of the pendulum. In addition, in double or multiple pendulum devices, significantly different length, flex and resonant operating characteristics of multiple pendulum arm flexures create substantial difficulties in attempting to coordinate and synchronize the motions of multiple pendulums, or may make synchronized operation achievable only when accompanied by substantial and undesirable energy loss.